Meusnier theorem - Encyclopedia of Mathematics (original) (raw)
If gamma\gammagamma is a curve lying on a surface and PPP is a point on gamma\gammagamma, then the curvature kkk of gamma\gammagamma at PPP, the curvature kNk_NkN of the normal section of the surface by the plane passing through both the unit tangent vector to gamma\gammagamma at PPP and the unit normal vector to the surface, and the angle alpha\alphaalpha between the referred plane of gamma\gammagamma at PPP and the osculating plane, satisfy the relation
kN=kcosalpha.k_N = k \cos \alpha .kN=kcosalpha.
In particular, the curvature of every inclined section of the surface can be expressed in terms of the curvature of the normal section with the same tangent.
This theorem was proved by J. Meusnier in 1779 (and was published in [1]).
References
| [1] | J. Meusnier, Mém. prés. par div. Etrangers. Acad. Sci. Paris , 10 (1785) pp. 477–510 |
|---|---|
| [a1] | M.P. Do Carmo, "Differential geometry of curves and surfaces" , Prentice-Hall (1976) pp. 142 |
| [a2] | W. Blaschke, K. Leichtweiss, "Elementare Differentialgeometrie" , Springer (1973) |
How to Cite This Entry:
Meusnier theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Meusnier\_theorem&oldid=53936
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article